Triplet Excitation Energy Transfer with Constrained Density Functional

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J. Phys. Chem. C 2010, 114, 20756–20763

Triplet Excitation Energy Transfer with Constrained Density Functional Theory† Sina Yeganeh and Troy Van Voorhis* Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307, United States ReceiVed: July 26, 2010; ReVised Manuscript ReceiVed: September 17, 2010

We study the electronic coupling matrix element for triplet excitation energy-transfer processes with a number of different computational methods. For the first time, constrained density functional theory (CDFT) is applied to the problem of energy transfer, and results are compared with direct coupling calculations of broken symmetry and fragment densities, as well as the splitting method. A naı¨ve calculation of the electronic coupling using diabatic and adiabatic energy differences is shown to yield erroneous results due to the fractional spin error present in both Hartree-Fock and commonly used DFT exchange-correlation functionals. Some potential issues concerning the splitting method with triplet references within Hartree-Fock and DFT are discussed. We find that only methods that compute the matrix element directly (either from CDFT, broken symmetry, or fragment states) appear to be robust. Several illustrative examples are presented. 1. Introduction Energy transfer is an important process in both natural and synthetic systems. For example, triplet excitation energy transfer (TEET) plays an important role in photosynthesis,1 where carotenoids prevent oxidation due to triplet-state chlorophyll via energy transfer.2 Studying TEET can also yield new insights into electron transfer as similar factors (exchange, overlap) govern both, and, in fact, TEET has been modeled as a double electron exchange reaction.3,4 In organic electronics,5 energy transfer is a key process in organic photovoltaic (OPV) and lightemitting diode (OLED) devices. Designing OLEDs requires, in particular, a clear understanding of TEET as spin statistics imply 75% of the created excitons will be phosphorescent rather than luminescent. In OPVs, some donor/acceptor materials have significant triplet exciton populations, and engineering greater fractions of triplet excitons might be a route to improving the short exciton diffusion lengths in these devices.6,7 Thus, TEET in a variety of different systems has been an active area of interest for experimentalists.8-12 Within the approximations inherent to Fermi’s golden rule (FGR), the nonadiabatic TEET rate can be written as

kTEET )

2π |V |2(DWFC) p if

(1)

where DWFC refers to the density-weighted Franck-Condon factors and Vif, the subject of this work, is the electronic coupling between initial and final states of the energy-transfer process. The Franck-Condon factors can be derived for classical13 or quantum harmonic oscillators,14 and multiple vibrational modes15 as well as anharmonic effects16,17 can be considered. These factors are important for a quantitative calculation of the reaction rate, but here, we are primarily concerned with deducing the distance dependence of energy transfer, which lies mainly in the electronic coupling. Although TEET is an overall spinconserving process, the Coulomb interaction, which leads to †

Part of the “Mark A. Ratner Festschrift”. * To whom correspondence should be addressed. E-mail: [email protected].

the usual R-3 distance dependence in Fo¨rster theory,18 is only weakly allowed by spin-orbit coupling and can be neglected.19 Therefore, TEET is primarily mediated by the remaining interaction terms, exchange and orbital overlap, resulting in exponential decay of the TEET rate as the donor and acceptor are separated.20-22 The electronic coupling for the energy transfer from excitedstate donor, D*, to ground-state acceptor, A, can be written as

ˆ |D*A〉 Vif ) 〈DA*|H

(2)

ˆ is the electronic Hamiltonian for the donor and acceptor where H and the interactions between them. The states |D*A〉 and |DA*〉 are diabatic-like, representing excitations localized on different molecules (intermolecular TEET) or different parts of the same molecule (intramolecular TEET). The identification of these diabatic states, either implicitly or explicitly, is the main task in calculating the electronic coupling. In this paper, we begin by describing the transformation from the adiabatic states of quantum chemistry to the diabatic states necessary for FGR-type rate expressions. We then describe how we identify the diabatic states for localized triplet excited states using constrained density functional theory (CDFT).23,24 With diabatic states in hand, we can calculate the electronic coupling directly. Using the diabatic-adiabatic energy gap provides an alternative approach; however, we find that this method fails due to the fractional spin error in single-reference methods, and we discuss the reasons for this failure as well as its effects on TEET calculations in terms of ensemble DFT. Some cautionary notes are also raised for care in selecting adiabatic states in the splitting method. Finally, we examine chemical systems of interest, from a simple system consisting of two formaldehyde chromophores to a larger intramolecular TEET system of porphyrin moieties studied experimentally by Albinsson and co-workers.11,25,26 2. Background 2.1. Diabatic-Adiabatic Transformations. We present a brief background to the subject of diabatic states and their

10.1021/jp106989t  2010 American Chemical Society Published on Web 10/05/2010

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construction from adiabatic states. For a more in-depth discussion, we refer to existing literature.24,27,28 Within the Born-Oppenheimer approximation, adiabatic states are those which remain an eigenstate of the electronic Hamiltonian as a slowly varying perturbation (e.g., change along some reaction coordinate) is applied. These are the familiar output of quantum chemical calculations where the ground and excited states are well-defined at all coordinates. On the other hand, a diabatic state is one which remains qualitatively similar in its electronic character even at vastly different nuclear configurations. For example, in an alkali halide compound, XY, the ground adiabatic state would change smoothly from being mainly ionic at close separations to covalent at the dissociative limit; on the other hand, the two diabatic states would remain similar at all internuclear separations, corresponding to X+Y- and XY. In computational quantum chemistry, the Born-Oppenheimer approximation is nearly always invoked, and adiabatic states are obtained. Although, in general, there is not a unique transformation to diabatic states,29 approximate diabatic states can be obtained via rotations of the adiabatic states. For two states, this transformation can be represented by the following generalized eigenvalue equation

(

)( ) (

)( )

εi Hif ci( 1 Sif ci( ) ε( Hif εf cf( Sif 1 cf(

(3)

where εi,f are the diabatic state energies, Hif/Sif are the diabatic state coupling/overlap, ci,f are the eigenvector coefficients from which the rotation angle between diabatic and adiabatic states can be obtained, and ε( are the two adiabatic state energies. In general, the diabatic states obtained will not be orthogonal; therefore, to obtain the electronic coupling matrix element required for the FGR expression (eq 1), a symmetric orthogonalization is carried out to yield

Vif )

Hif - Sif(εi + εf)/2 1 - Sif2

(4)

We can solve the eigenvalue equation and examine the result in a number of cases. If there is some symmetry in the system (such as identical monomers in an intermolecular TEET) such that εi ) εf, the coupling can be written

Vif )

ε- - ε+ 2

(5)

and this result is commonly referred to as the splitting method. For symmetric systems, this result tells us that simply calculating the two appropriate adiabatic states and taking half of their energy splitting yields the electronic coupling. Generally, a singlet ground-state reference is used, and CIS, TDHF, or TDDFT (within the Tamm-Dancoff (TDA) or random phase (RPA) approximations) are used to obtain the T1 and T2 energies. Taking half of the difference between these two values yields the desired coupling. In another limiting case, when the overlap between diabatic states is negligible (Sif ≈ 0), the coupling reduces to

Vif ) √(εi - ε+)(εf - ε+)

(6)

We see that only one adiabatic state (T1) is required in addition to the two diabatic state energies; in the case of symmetric diabats and negligible overlap, Vif ) εi - ε+. Finally, some algebra yields the coupling in the general case

Vif )

√(εi - ε+)(εf - ε+) + Sif(ε+ - (εi + εf)/2) 1 - Sif2

(7) where Sif can be obtained by direct calculation of the overlap between the two diabatic states or through the analytic expression

Sif )

√(εi - ε+)(εf - ε+) + √(εi - ε-)(εf - ε-) ε- - ε+

(8) An alternative path to Vif can be found by directly obtaining diabatic states and explicitly calculating the coupling. One method for obtaining such diabatic states for TEET is CDFT, as discussed next. 2.2. Constrained Density Functional Theory. We introduce CDFT23 as a constructive method to yield diabatic states for calculating TEET rates. More in-depth discussion can be found in the literature.24 In CDFT, an appropriately chosen constraint is added to the usual energy functional, E[F], forcing the ground state of the constrained system to converge to the desired diabatic state. The constraint is enforced during SCF optimization via a Lagrange multiplier, Vc. The variational principle is then applied to W[F]

W[F] ) E[F] + Vc

(

R,β

)

∑ ∫ dr wσc (r)Fσ(r) - Nc σ

(9)

where Nc is the desired constraint value and wc serves as the weight function that determines the constraint. Here, we make use of Becke’s atomic partitioning scheme30 to define wc. By setting wRc * wβc , spin constraints can be defined; setting R wc ) 1 and wβc ) -1 in a region where the excitation is to be localized, and zero elsewhere, and concurrently setting Nc to 2 will result in a localized triplet. With an appropriate choice of constraint regions, this yields the desired diabatic states for describing TEET (see Figure 1). One additional note should be made regarding the choice of the constraint value Nc. For an intermolecular charge or spin-localized state where the fragments are considerably separated, integer choice of Nc is appropriate; however, when the fragments are close together or in an intramolecular localized state, strictly integer values of Nc are no longer appropriate, and a promolecule approach is needed to obtain Nc.24,31,32 For our calculations here, we generally make use of this promolecule correction, although the coupling results do not change significantly in most cases. A development version of Q-Chem was used for our calculations.33 Finally, we note that since HF can be formulated as an exchange-correlation functional within generalized Kohn-Sham (KS) DFT, the CDFT approach outlined above can readily become a constrained HF (CHF) method. With diabatic states from CDFT, we can now compute the diabatic coupling. The most obvious approach is to directly evaluate the nonorthogonal coupling with CDFT providing the |DA*〉 and |D*A〉 states. While this prescription is appropriate

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Figure 1. Spin density (FR-Fβ) is plotted for the naphthalene dimer in adiabatic (center) and diabatic (left/right) representations. Methods for computing the electronic coupling either obtain adiabatic states and indirectly calculate the coupling or (as in CDFT) directly obtain diabatic states and evaluate the coupling matrix element between them.

for CHF, in CDFT, it requires evaluating one- and two-electron matrix elements over diabatic wave functions constructed from KS orbitals and is thus questionable. An alternative approach is to rewrite the matrix element to avoid requiring two-body matrix elements24,34

ˆ + VDc *Awc - VDc *Awc |D*A〉 Hif ) 〈DA*|H ) (E +

* VDc ANc)Sif

-

* VDc A〈DA*|wc |D*A〉

(10) where we have reduced the coupling dependence on the KS orbitals to a zero-electron overlap term, Sif, and the one-body matrix element over the weight function. We refer to this method as the CDFT coupling. Finally, from the discussion on diabatic-adiabatic transformations in section 2.1, we saw in eq 7 that the diabatic coupling could be calculated from two diabatic and one adiabatic state energies, without any explicit calculation of the coupling matrix element between the diabatic states. Using CHF/CDFT to obtain εi,f and Sif while obtaining the adiabatic energy, ε+, from the ground-state triplet energy of the system, we should be able to obtain an alternative estimate of the electronic coupling. We term this method CDFT-∆. As we find, however, the requirement of accurate adiabatic state energies causes difficulties in applying this method.34 2.3. Fractional Spin. Fractional charge error has been widely discussed in the context of DFT. The key point is that the selfinteraction error (SIE) in commonly used exchange-correlation functionals causes an excessive delocalization of the electronic density, leading to, among other errors, too low barriers, too large polarizabilities, and overestimation of conductance in molecular electronics calculations.35 An alternative view of this delocalization error36 is based on the ensemble DFT proof that the true functional should yield a linear dependence of the total energy on the electron number between adjacent integer values,37 with the slope of E(N) being proportional to either the ionization potential or electron affinity on either side of the integer electron number. SIE, however, causes approximate functionals to display nonlinear convex dependence on fractional charge, resulting in a too low, overly delocalized description of systems. As system size increases, this delocalization error grows. Although HF does not possess a similar SIE, energies have an incorrect concave dependence on electron number, resulting in a localization error, which saturates with increasing system size.35 A closely related, although less discussed, issue is fractional spin error.38 Consider a hydrogen atom, usually pictured as containing one spin-up (or spin-down) electron. Next, imagine a hydrogen atom which contains half of a spin-up and half of a spin-down electron. The exact functional should give the same

Figure 2. A sketch of the fractional spin error in atomic hydrogen. While the exact functional should yield a constant energy for all configurations of partial up and down spin, both HF and LDA display concave behavior due to the single determinant description in both methods.

energy for all of these configurations.38 However, approximate functionals (and HF) all yield large deviations from this exact condition. This failure results from the inability of mean-field HF and DFT functionals to describe the multiconfigurational nature of the half-up half-down state. Thus fractional spin error can be regarded as a static correlation error which arises when describing these degenerate states. In previous studies, the fractional spin error for both HF and DFT functionals has resulted in improper concave behavior (see Figure 2), although, as we will see later, this is not always the case in molecular systems. For our interests in TEET, we will be concerned with a slightly different form of the fractional spin error; we will study delocalized spin across a dimer. Locally, these delocalized solutions will look like they have fractional spin; each monomer will be approximately 50% singlet and 50% triplet in character. On the other hand, the localized solutions will correspond to the end points, with one monomer entirely singlet and the other triplet. Thus, the extreme points in Figure 2 correspond to all singlet or all triplet, with intermediate points a mixture of the two configurations. Regardless of whether the midpoint or endpoints are taken, the energy of the system should be the same. However, this is not the case in HF or DFT, and thus, the energy of a delocalized spin configuration is expected to be incorrect. Thus, we expect that adiabatic triplet-state energies for dimer systems will be of questionable accuracy, and electronic coupling methods that rely upon these states and their energies will likewise be flawed. As TEET couplings are generally small, it is not surprising that this fractional spin error will be calamitous for the CDFT-∆ method. Additionally, the incorrect concavity in Figure 2 suggests that linear response calculations from a triplet-reference dimer system will be in error. 3. Results 3.1. Coupling Methods. Before discussing the results of TEET coupling calculations with CDFT, we introduce some of the methods that we use for comparison. The different methods can be separated into two categories based on the way that they construct the diabatic coupling and the reference states used.

Triplet Excitation Energy Transfer with Constrained DFT In the first category, adiabatic-based methods, the singlet groundstate reference is usually used, and information about the triplet excited states is obtained through linear response. The simplest, and most often used, example is the splitting method, applicable to systems with symmetry conditions such that the two diabatic energies are the same. For TEET, generally CIS/TDDFT-TDA or TDHF/TDDFT-RPA calculations are performed with the singlet reference, and the two lowest triplet-state energies yield the coupling via eq 5. Alternatively, post-HF excited-state methods (such as equation-of-motion coupled cluster) can be used for more accurate adiabatic energies. This method, in addition to the necessity of symmetry, also requires an appropriate selection of adiabatic states; for this reason, HF, rather than DFT, is often used as TDDFT can yield spurious lowlying charge-transfer states in dimer calculations, which would result in incorrect splitting values of the coupling.39-41 A new method, fragment excitation difference (FED),42,43 can also be used for TEET, even in asymmetric systems. It is based on similar ideas as the fragment charge difference method for electron-transfer coupling elements,44 and like the splitting method, it relies on accurate adiabatic state energies as well as transition densities from CIS/TDDFT. As in CDFT, it requires the specification of the donor and acceptor regions. We note that for splitting and FED, an alternative approach would be to begin with the triplet ground state and perform excited-state calculations to the first excited state; this approach fails however, yielding drastically overestimated TEET couplings. We discuss the source of this behavior further in section 3.2. CDFT-∆ can also roughly be placed in this category as it makes use of both adiabatic and diabatic energies. The second category consists of direct coupling methods with explicit calculation of the triplet state, and the CDFT coupling method falls into this category. In addition, we will compare our CDFT results with the broken symmetry (BS) method.45 Here, approximate diabatic states are obtained by forcing the self-consistent field iterations to converge to a UHF/UKS state with the excess spin localized on one fragment. This is usually readily done in HF calculations but can be considerably more challenging in pure DFT calculations, where the system often converges to the delocalized solution. Once these diabatic states are obtained, the electronic coupling is obtained by evaluating

ˆ |D*A〉 Hif ) 〈DA*|H

(11)

Sif ) 〈DA*|D*A〉

(12)

and applying eq 4 to obtain Vif for orthogonal states. An approximate and computationally cheap alternative for obtaining the TEET coupling is the fragment method. Here, a diabatic state is obtained by forming the block diagonal density matrix from the combination of two isolated monomer calculations, one monomer in its singlet ground state and the other in its triplet ground state. The coupling is once again obtained from eq 11. As this requires separate calculations of the donor/ acceptor fragments, this method is best suited for intermolecular TEET. 3.2. TEET in Stacked Formaldehyde and Naphthalene. We now turn to benchmarking our CDFT method. As a first simple example, we study a stacked formaldehyde dimer. The electronic coupling matrix element for TEET from one formaldehyde to another in a parallel stacked configuration is calculated while varying the separation distance. From Dexter theory, we expect an exponential distance dependence of this

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Figure 3. The electronic coupling for TEET in stacked formaldehyde is shown with a number of different methods (6-31G* basis). For HF (a), the broken symmetry method, CHF coupling, and HF splitting are shown. For PBE (b), CDFT-∆, CDFT coupling, and splitting are shown; broken symmetry was inapplicable as the diabatic states could not be obtained. The absolute value of the coupling is plotted; hence, the discontinuity in the CDFT-∆ result corresponds to a change in sign.

coupling, with the decay of Vif equal to β/2, where β is the usual attenuation factor for exponentially decaying rates. For the CHF/CDFT-∆ method, truly adiabatic states were obtained by enforcing symmetry in the SCF convergence. HF and DFT (PBE functional)46 calculations were carried out with the 6-31G* basis. In Figure 3, we see that for HF, the splitting, BS, and CHF coupling methods all decay exponentially with similar β values. For PBE, we see similar β for the CDFT coupling and splitting method (BS states could not be obtained in PBE). For both HF and PBE, however, the CHF/CDFT-∆ result is qualitatively different, yielding a different slope and nonzero coupling at infinite distances! The CHF- ∆ result (not shown) was much larger than those from the other methods, yielding a coupling that grew from 3000 to 5000 meV as the distanced increased. As the ∆ calculation uses the same diabatic state as that in direct coupling CHF/CDFT, the adiabatic state energy seems to be the culprit. Additionally, the splitting method was applied to calculations from the triplet reference (not shown); we find a coupling that is several orders of magnitude too large (∼1000 meV) and nearly constant with separation. Although the CHF/CDFT, BS, and splitting methods all yield similar slopes for the coupling, the absolute magnitudes differ. The difference between the diabatic- and adiabatic-based schemes can be explained by the fact that diabats cannot be uniquely obtained from adiabats, and using the two lowest-lying adiabatic states does not guarantee an adequate basis for the transformation. Comparing different diabatic schemes, such as CDFT and BS, we find that the choice of diabatization procedure can affect the TEET coupling.

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Figure 4. The TEET electronic coupling for a pair of π-stacked naphthalene molecules is shown with a number of different methods for HF (a) and PBE (b) (6-31++G** basis). The CHF/CDFT coupling, splitting, and fragment method results are plotted.

Next, we turn to the π-stacked naphthalene dimer, and using both HF and PBE (6-31++G**), we calculate the TEET coupling with CHF/CDFT coupling and ∆, as well as the splitting and fragment methods. As shown in Figure 4, we find that CHF/CDFT, splitting, and fragment methods all yield exponential decay, with similar slopes. Once again, the CHF/ CDFT-∆ result is much too large and qualitatively incorrect (not shown), with an anomalous rise in the coupling as a function of distance for PBE. The magnitude of the TEET coupling is considerably larger than that in the formaldehyde dimer, as expected for the more delocalized, π-conjugated naphthalene system. For both formaldehyde and naphthalene, as well as other test systems not discussed here, we found that the CHF/CDFT-∆ method yielded qualitatively incorrect (nonexponential, nonvanishing) coupling. The origin of this error is the adiabatic state energy, that is, the ground-state triplet energy calculated with either HF or DFT functionals. For a symmetric system with infinite donor-acceptor separation, the energy of the diabatic state (with the triplet localized on one site) should exactly equal the energy of the ground adiabatic state (with the triplet delocalized over both), leading to zero coupling as expected. This is a consequence of the constancy condition for fractional spin,38 discussed earlier for the hydrogen atom. Here, the constancy condition for a single molecule can be written for an electron density, F, as

respectively. The energy of this ensemble density should vary linearly as a function of γ, with E[F(1)] - E[F(0)] equal to the singlet-triplet gap for the isolated molecule. As we will see, this is not generally the case with HF and DFT functionals. In Figure 5, E[F(γ)] is calculated for formaldehyde, naphthalene, and lithium fluoride with the 6-311+G** basis and a number of different exchange-correlation functionals. To aid the eye, the linear change in energy from the singlet to triplet expected from the exact condition is differenced to yield a symmetric curve, so that the exact result should be a flat line; in this way, deviations from constant energy are more easily seen. Thus, we plot

F(γ) ) γFS0 + (1 - γ)FT1

j [F(γ)] ) E[F(γ)] - γ(E[F(1)] - E[F(0)]) E

(13)

where γ:[0, 1] represents the fraction of singlet density, and FS0 and FT1 are the ground-state singlet and triplet densities,

Figure 5. The ensemble density energy, eq 14, is plotted for (a) formaldehyde, (b) naphthalene, and (c) lithium fluoride. Deviation from j [γ] ) 0, is evidence of fractional spin error. All calculations a flat line, E were performed with the 6-311+G** basis.

(14)

In Figure 5a,b, for formaldehyde and naphthalene, we find that HF has the largest fractional spin error and incorrect concavity,

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J. Phys. Chem. C, Vol. 114, No. 48, 2010 20761 TABLE 1: Extrapolated β Values for the Biporphyrin System in Figure 6

Figure 6. Structure of the biporphyrin series studied here. The experimental system (ref 11) had side-chains which are omitted in our calculation. The experimental system has X ) Zn and Y ) H2. The number of bridge units, n, is varied from 0 to 3.

and LDA has much smaller (but still slight) concavity; these trends are the same as those seen previously for atomic and molecular systems.38 The other functionals lie between these two, with hybrid functionals having larger deviations than the pure PBE functional. In Figure 5c for LiF, however, we find that while HF remains concave, the DFT functionals have an opposite error with incorrect convex behavior. The results in Figure 5 suggest that the adiabatic triplet energy in most cases will be too high, and eq 7 will yield too large of a coupling. The magnitude of this error at infinite separation j [F(1)/(2)] - E j [F(0)], and this error can be approximated from E estimate is in qualitative agreement with the overestimation of the coupling from the CHF/CDFT-∆ method. For LiF, we expect that the DFT adiabatic triplet energy will be too low in energy, and the coupling will once again be too large (but with opposite sign). The behavior described above might also explain the failing of the splitting method when applied to calculations from the triplet reference, as discussed briefly in section 3.1. In a dimer system, a TDHF or TDDFT calculation from the triplet ground state should yield a low-lying spin-conserving excitation, the magnitude of which should approach zero exponentially as the molecules are separated. In all calculations that we attempted, however, this behavior was not seen. Instead, the lowest-lying triplet excitation was much larger than expected and stayed roughly constant as separation was increased. This can be explained from Figure 5; within the linear response formulation47 of TDDFT, the RPA equations yield the excited-state energies of the system as the eigenvalues of a generalized eigenvalue equation for a matrix Γ. This Γ matrix can be shown48 to operate as the second derivative of the energy with respect to orbital rotations, (∂2E)/(∂cia∂cjb). For systems with concave behavior (such as formaldehyde and naphthalene), a linear response calculation from the delocalized adiabatic state, F(1/2), will fail, resulting in imaginary roots. Thus, a TDHF/TDDFT calculation from the triplet reference requires that the partial derivative be evaluated at the left/right extremes of the plot (γ ) 0, 1). At these points, the second derivative for the exact functional should be zero, yielding a doubly degenerate triplet ground state at infinite separation; for approximate functionals, however, the second derivative can be quite large due to incorrect concavity, and too large excitation energies result. It has been pointed out previously that SIE and fractional charge error make the CDFT-∆ method inapplicable for calculating the electronic coupling between charge-transfer diabats.49 From our calculations, we conclude that a similar failing of the exact constancy conditions for fractional spin leads to errors when using the CHF/CDFT-∆ method for TEET as well. 3.3. Intramolecular TEET in Biporphyrins. Finally, as a more chemically interesting example, we examine a series of biporphyrin molecules studied experimentally and theoretically by Eng et al.11 As shown in Figure 6, the separation between a free base (H2P) and zinc (ZnP) porphyrin was increased by

β (Å-1)

splittinga

CDFT

expt.a

H2-H2 Zn-Zn Zn-H2

0.32 0.19

0.53 0.51 0.50

0.45

a Calculated splitting and experimental values from ref. 11 are also shown.

Figure 7. The TEET electronic coupling for the biporphyrin systems in Figure 6 is plotted. Calculations with CDFT (6-31G*, B3LYP) are shown. Calculated β values are found in Table 1.

adding oligophenylene ethynylene units. The rate of TEET was measured from transient absorption studies, and an attenuation factor of β ) 0.45 Å-1 was found. In addition, TDDFT calculations (6-31G*, B3LYP) were carried out, and the splitting method was used to estimate β; to make the splitting method applicable, symmetric analogues of the experimental system were calculated with H2P-H2P, yielding β ) 0.32 Å-1, while ZnP-ZnP was found to have β ) 0.19 Å-1, both with a planar bridge geometry. Using Boltzmann-weighted geometries rather than assuming planarity yielded a slightly larger value for β in each case. We perform the TEET coupling calculation for the asymmetric system, as well as the symmetric analogues, with CDFT. After geometry optimization (with symmetry constraints forcing the entire molecule to remain planar), we calculated the localized triplet diabatic states and the coupling with our CDFT method (6-31G*, B3LYP). The triplet spin was constrained to be on one-half of the molecule (porphyrin and half of the bridge) or the other. Although our model system is not identical to the experimental molecules (which had β-ethyl groups on the porphyrin rings, resulting in a large dihedral angle between porphyrins and the bridge), we expect that the distance dependence of the electronic coupling should be fairly similar, while the absolute magnitude of the coupling could be significantly larger than that with the twisted bridge. Our results are summarized in Table 1 and plotted in Figure 7. For the mixed system, we find β ) 0.50 Å-1, in good agreement with the experimental result and recently published theoretical results with the fragment spin difference method (FSD).43 In contrast with the splitting result, we find that the substituents in the porphyrins (Zn or H2) have little effect on the distance dependence of the electronic coupling (or its overall magnitude). 4. Conclusions In the preceding, we have discussed the various methods available for calculation of the electronic coupling in energy transfer of triplet excited states. These can be divided roughly

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into two classes, those based on energy gap expressions for the coupling and those based on the triplet state and direct coupling from diabatic states. In the former, the splitting method for symmetric systems and FED provide a means of obtaining the electronic coupling for triplet transfer via linear response calculations of the triplet excited states. These methods thus require accurate values of the triplet energy levels. Although not discussed here, we note recent work which has found that TDDFT errors for triplet excited states are in general larger than those for singlets.50 In addition to quantitative errors in the triplet excitation spectrum, there is also the potential for the lowest calculated triplet excited states to be qualitatively incorrect due to artificially low-lying charge-transfer39-41 or other intruder states.51 If these adiabatic states are chosen, the diabaticadiabatic transformation will fail to yield the appropriate electronic coupling. Another class of methods for computing electronic coupling is based on defining some additional operator which represents the degree of excitation-localization; by maximizing this operator, the rotation matrix from the computed adiabatic states to the desired diabatic states is derived. The FED and FSD methods can also be classified in this way. Additionally, a recently proposed approach by Subotnik et al. makes use of orbital localization techniques to localize states and can yield electron- and energy-transfer couplings.52-54 These methods hold significant promise for extending the splitting method to asymmetric systems while only requiring adiabatic states. However, they will still be plagued by the same difficulties as discussed above; in single-reference methods, the triplet adiabatic state will be poorly described, and there remains an ambiguity as to the choice of correct adiabatic basis for the transformation to diabatic states. The second class of electronic coupling methods discussed in this work is based on calculating triplet diabatic states directly. Both perturbative (fragment) and supramolecular (CDFT and broken symmetry) approaches were discussed. We proposed a new approach for TEET coupling, where CDFT provides the necessary diabatic states, and the expression in eq 10 is used to obtain the coupling. Unlike other direct coupling methods, CDFT can be applied to both inter- and intramolecular TEET and has a robust optimization method for converging to the diabatic state. Direct construction of diabatic states avoids ambiguity involved with the choice of the correct adiabatic basis, and in addition, directly evaluating the coupling matrix element from the CDFT diabats obviates any potential issues from fractional spin error. Compared to other methods which evaluate the electronic coupling from diabatic states, CDFT has a number of advantages. The broken symmetry method, while similar to CDFT, is not robust, and it is often difficult to force the calculation to converge to the desired states. The fragment method can only be applied to intermolecular TEET and neglects the indirect effect of interactions between the fragments on the electronic structure of the dimer complex. Finally, we found that the fractional spin error present in HF and commonly used DFT functionals can contaminate the coupling in triplet reference calculations if one is not careful. The adiabatic triplet-state energy is incorrect, and even small deviations result in relatively large errors in the coupling. Thus, we conclude that direct coupling schemes, in which the electronic coupling is calculated by directly evaluating the matrix element between diabats, are a more robust choice for TEET. Looking forward, there are a number of related avenues of interest. One important question is what the best diabatic method for FGR-type rate expressions is; although there is not generally

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